XI] ITS GENERAL PROPERTIES 501 



let Fig. 242 represent in section the early growth of a Nautilus- 

 shell, and let the part ARB represent the earUest stage of all, 

 which in Nautilus is nearly semicircular. We have to find a law 

 governing the growth of the shell, such that each edge shall 

 develope into an equiangular spiral; and this law, accordingly, 

 must be the same for each edge, namely that at each instant the 

 direction of growth makes a constant angle with a Hne drawn from 

 a fixed point (called the pole of the spiral) to the point at which 

 growth is taking place. This growth, we now find, may be 

 considered as effected by the continuous addition of similar 

 quadrilaterals. Thus, in Fig. 241, AEDB is a quadrilateral with 

 AE, DB parallel, and with the angle EAB of a certain definite 



Fig. 242. 



magnitude, = y. Let AB and ED meet, when produced, in C ; 

 and call the angle ACE (or xCy) = j8. Make the angle yCz = angle 

 xCy, = j8. Draw EG, so that the angle yEG = y, meeting Cz in 

 G\ and draw BE parallel to EG. It is then easy to show that 

 AEDB and EGFD are similar quadrilaterals. And, when we 

 consider the quadrilateral AEDB as having infinitesimal sides, 

 AE and BD, the angle y tends to a, the constant angle of an equi- 

 angular spiral which passes through the points AEG, and of a 

 similar spiral which passes through the points BDF ; and the point 

 C is the pole of both of these spirals. In a particular hmiting case, 

 when our quadrilaterals are all equal as well as similar, — which 

 will be the case when the angle y (or the angles EAC, etc.) is a 



